Please can someone tell me whether a player can legally
resign at any point in a game? Can the opponent insist
that play continues to see whether the final result is a
gammon or backgammon? Or impose a condition, saying that
the resignation is accepted provided the loser concedes a
gammon or backgammon (and not an ordinary game)?
Many thanks in advance.
Bob

A player may offer to resign (or settle) at any time. (Online, a player
may resign only when it is their turn.) Their opponent is free to accept
or decline the offer, or make a counteroffer. Sometimes, a player will try
to "game" you into accepting less than full value, but this little ploy is
easily refused -- smile brightly, and counteroffer.
Technically, either side can insist on continuation until one side has all
their checkers off the board. However ...
Let's consider money play first. Every point matters to the bottom line.
Unless the offer is to your advantage (in which case you should just gobble
it up), you should insist that play continue so long as the outcome is in
doubt. The usual way to think about this is: If you could call the rolls
for both sides (for example, all 66 for you, all 21 for your opponent),
would the outcome change from what has been offered? Then make the same
calculation from your opponent's side. If the outcome would not change
from that offered, then there is no practical point in continuing, and you
should accept the offer and move on to the next game. (As Kit Woolsey has
observed, it may be faster to roll it out than try to figure it out. The
adage that "time is money" applies here.) Of course, there are sometimes
psychological considerations, but that is a different issue.
Now let's consider standard match play (used in most tournaments). Extra
points above those required to win the match do not matter. You derive no
benefit from the actual match scores, but only in winning or losing each
match. If your opponent offers sufficient points for you to win the match,
it is considered bad sportsmanship to insist that your opponent continue
"just to see if I can win a gammon." A win is a win. You should accept
graciously, record your victory, and move on to your next match. Of
course, if the actual scores matter, then you should make the decision to
demand continuation accordingly.
Inexperienced match players sometimes fall into this error, insisting "I
could win a gammon" when the point(s) offered win the match. When you
encounter this situation, it is worth trying to explain match scoring, as a
teacher would to a student. On the other hand, if the actual score doesn't
matter, you can expedite the ending by offering more than is required. You
lose nothing of value, and you can take comfort in your superior knowledge
of match equity.
Regards,
Richard